Advertisement

Israel Journal of Mathematics

, Volume 125, Issue 1, pp 93–130 | Cite as

Gibbs states on the symbolic space over an infinite alphabet

  • R. Daniel Mauldin
  • Mariusz Urbański
Article

Abstract

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Hölder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Keywords

Variational Principle Incidence Matrix Iterate Function System Borel Probability Measure Gibbs State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ar]
    J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs 50, American Mathematical Society, Providence, RI, 1997.Google Scholar
  2. [AD]
    J. Aaronson and M. Denker,Local limit theorems for Gibbs-Markov maps, Preprint, 1996.Google Scholar
  3. [ADU]
    J. Aaronson, M. Denker and M. Urbański,Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of the American Mathematical Society337 (1993), 495–548.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Bo]
    R. Bowen,Equlibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics470, Springer-Verlag, Berlin, 1975.Google Scholar
  5. [DKU]
    M. Denker, G. Keller and M. Urbański,On the uniqueness of equilibrium states for piecewise monotone maps, Studia Mathematica97 (1990), 27–36.zbMATHMathSciNetGoogle Scholar
  6. [DU1]
    M. Denker and M. Urbański,Relating Hausdorff measures and harmonic measures on parabolic Jordan curves, Journal für die Reine und Angewandte Mathematik450 (1994), 181–201.zbMATHGoogle Scholar
  7. [Gu]
    B. M. Gurevich,Shift entropy and Markov measures in the path space of a denumerable graph, Doklady Akademii Nauk SSSR192 (1970); English transl.: Soviet Mathematics Doklady11 (1970), 744–747.Google Scholar
  8. [GS]
    B. M. Gurewic and S. V. Savchenko,Thermodynamic formalism for countable symbolic Markov chains, Russian Mathematical Surveys53 (1998), 245–344.CrossRefGoogle Scholar
  9. [HMU]
    P. Hanus, D. Mauldin and M. Urbański,Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Preprint, IHES, 1999.Google Scholar
  10. [HU]
    P. Hanus and M. Urbański,A new class of positive recurrent functions, Contemporary Mathematics246 (1999), 123–136.Google Scholar
  11. [IL]
    I. A. Ibragimov and Y. V. Linnik,Independent and stationary sequences of random variables, Walters-Noordhoff Publ., Groningen, 1971.zbMATHGoogle Scholar
  12. [ITM]
    C. Ionescu-Tulcea and G. Marinescu,Théorie ergodique pour des classes d’operations non-complement continues, Annals of Mathematics52 (1950), 140–147.CrossRefMathSciNetGoogle Scholar
  13. [MU1]
    D. Mauldin and M. Urbański,Dimensions and measures in infinite iterated function systems, Proceedings of the London Mathematical Society (3)73 (1996), 105–154.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [MU2]
    D. Mauldin and M. Urbański,Conformal iterated function systems with applications to the geometry of continued fractions, Transactions of the American Mathematical Society351 (1999), 4995–5025.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [MU3]
    D. Mauldin and M. Urbański,Parabolic iterated function systems, Ergodic Theory and Dynamical Systems20 (2000), 1423–1447.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Or]
    D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, New Haven and London, 1974.zbMATHGoogle Scholar
  17. [PP]
    Y. Pesin and B. Pitskel,Topological pressure and variational principle for noncompact sets, Functional Analysis and its Applications18 (1985), 307–318.CrossRefGoogle Scholar
  18. [PS]
    W. Philipp and W. Stout,Almost sure invariance principles for partial sums of weakly dependent random variables, Memoirs of the American Mathematical Society161 (2) (1975).Google Scholar
  19. [PU]
    F. Przytycki and M. Urbański,Fractals in the Plane—Ergodic Theory Methods, Cambridge University Press, to appear; available on Urbański’s webpage.Google Scholar
  20. [Ru]
    D. Ruelle,Thermodynamic formalism, Encyclopedia of Mathematics and its Applications 5, Addison-Wesley, Mass., 1978.Google Scholar
  21. [Ry]
    M. Rychlik,Bounded variation and invariant measures, Studia Mathematica76 (1983), 69–80.zbMATHMathSciNetGoogle Scholar
  22. [Sa]
    O. Sarig,Theormodynamic formalism for countable Markov shifts, Ergodic Theory and Dynamical Systems19 (1999), 1565–1593.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Ur]
    M. Urbański,Hausdorff measures versus equilibrium states of conformal infinite iterated function systems, Periodica Mathematica Hungarica37 (1998), 153–205.CrossRefMathSciNetGoogle Scholar
  24. [Wa]
    P. Walters,An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982.zbMATHGoogle Scholar
  25. [Za]
    A. Zargaryan,Variational principles for topological pressure in the case of Markov chain with a countable number of states, Mathematical Notes of the Academy of Sciences of the USSR40 (1986), 921–928.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2001

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North TexasDentonUSA

Personalised recommendations